Affine network model, rubber elasticity

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

This work is based on the molecular dynamic simulation of a monomer scale model corresponding to the affine network model of rubber elasticity.3 However, whereas the classic model has no nonbonded interactions, our model... 

The Theory of Kuhn and Grun. The theory of birefringence of deformed elastomeric networks was developed by Kuhn and Griin and by Treloar on the basis of the same procedure as that used for the development of the classical theories of rubber-like elasticity (48,49). The pioneering theory of Kuhn and Griin is based on the affine network model that is, upon the application of a macroscopic deformation the components of the end-to-end vector for each network chain are assumed to change in the same ratio as that of the corresponding dimensions of the macroscopic sample. 

Rubber and other pol>nneric materials are elastic. Polymeric elastomers are covalently cross-linked networks of pol>Tner chains. Here w e describe one of the simplest and earliest models for the retractive forces of polymeric materials, the affine network model. 

Modern Theories, The term modern refers to theories of rubber-like elasticity introduced after 1975 mainly to account for the disagreement between experiment and the predictions of the phantom or affine network models. All... 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

The classical rubber elasticity model considers, however, that the crosslink points are particular, such that the cut-off occurs by these points in real space. The corresponding calculations for a chain obliged to pass by several crosslinks are recalled in Ref The calculation for the junction affine model was accomplished by Ull-mann for R and by Bastide for the entire form factor for the case of the phantom network model, this was achieved by Edwards and Warner using the replica method. 

Recent experimental evidence using SANS instrumentation suggests that the ends of a network segment deform affinely, yet the chain itself barely extends in the direction of the stress and contracts in the transverse direction even less. Develop a model to explain the results, and comment on how you think the theory of rubber elasticity ought to be modified to accommodate the new finding. 

---